OFFSET
0,2
COMMENTS
The denominators are given in A225149.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
FORMULA
a(n) = numerator((-1)^n*sum(l=0..2*n, 2^l * (sum(k=0..l, (k!*stirling2(l,k) * stirling1(l+k,l)) / (l+k)!)) * binomial(2*n,l))).
a(n) = numerator(b(n)), where b(n) = (-1)^n*(1-1/(2*n+1)-sum(i=1..n-1, b(i)*(-1)^i/(2*(n-i)+1))), b(0)=1. [Vladimir Kruchinin, Aug 29 2013]
EXAMPLE
x/((x^2+1)*atan(x)) = 1 - 2/3*x^2 + 26/45*x^4 - 502/945*x^6 + 7102/14175*x^8 - 44834/93555*x^10 + 295272982/638512875*x^12 - 122850554/273648375*x^14 + ...
MATHEMATICA
a[n_] := (-1)^n*Sum[2^l*(Sum[(k!*StirlingS2[l, k]*StirlingS1[l+k, l])/(l+k)!, {k, 0, l}])* Binomial[2*n, l], {l, 0, 2*n}]; Table[a[n] // Numerator, {n, 0, 14}] (* Jean-François Alcover, Apr 30 2013, translated from Maxima *)
Take[CoefficientList[Series[x/((x^2+1)ArcTan[x]), {x, 0, 30}], x], {1, -1, 2}]//Numerator (* Harvey P. Dale, Dec 26 2019 *)
PROG
(Maxima) a(n):=(-1)^n*sum(2^l*(sum((k!*stirling2(l, k) * stirling1(l+k, l))/(l+k)!, k, 0, l)) * binomial(2*n, l), l, 0, 2*n).
(PARI) x='x+O('x^66); v=Vec(x/((x^2+1)*atan(x))); vector(#v\2, n, numerator(v[2*n-1])) \\ Joerg Arndt, Apr 29 2013
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Vladimir Kruchinin, Mar 15 2013
STATUS
approved